Existence and Uniqueness of the Smallest Convex Superset of a Subset of $\mathbb{R}^n$

In the setting of Convex Sets in $\mathbb{R}^n$: Standing Notation, let $S\subseteq\mathbb{R}^{n}$.

(i) Existence and uniqueness. There is exactly one convex subset $H$ of $\mathbb{R}^{n}$ such that $S\subseteq H$ and $H\subseteq C$ for every convex $C\subseteq\mathbb{R}^{n}$ with $S\subseteq C$.

(ii) Description. This set $H$ is the intersection of all convex subsets of $\mathbb{R}^{n}$ containing $S$:

$$H=\{x\in\mathbb{R}^{n}:\ x\in C\text{ for every convex }C\subseteq\mathbb{R}^{n}\text{ with }S\subseteq C\}.$$